1366
Macromolecules 1986,19, 1366-1374
by the temperature change of a few degrees near T , of the liquid crystal. The viscosity of the nematic phase of EBBA is in the same range as that of liquid water, and the liquid-crystalline domain supposedly acts as the efficient mobile region of the gases. It was concluded by these authors that the temperature dependence of P is influenced by the thermal motion of the membrane component as well as by the continuity and/or size of the liquid-crystal phase. An analogous argument may be presented for interpretation of the permeation characteristics of the multicomponent bilayer film. Discontinuous jump of P a t T, is ascribable to greater gas mobility in the liquid-crystalline phase of the hydrocarbon component, and the jump becomes larger as the hydrocarbon domain is enlarged. Concluding Remarks Multicomponent bilayer membranes can be immobilized in the form of PVA composite films. The DSC and XPS data indicate that the hydrocarbon and fluorocarbon bilayer components are phase-separated and that the fluorocarbon component is concentrated near the film surface. These component distributions produce favorable effects on permselectivity of O2 gas. The selectivity (Po,/PN,)is apparently determined by the surface monolayer (or layers close to the surface) of the fluorocarbon component, and the permeability is promoted by the presence of large domains of the fluid (in the liquid-crystalline state) hydrocarbon bilayer.
Acknowledgment. We extend our appreciation to the Asahi Glass Foundation for Industrial Technology for financial support. Registry No. 1,91362-66-2;2, 89373-65-9;3, 100993-84-8;4, 100993-85-9;5 , 82838-66-2;02,7782-44-7;N2, 7727-37-9. References a n d Notes Nakashima, N.; Ando, R.; Kunitake, T. Chem. Lett. 1983,1577. Shimomura, M.; Kunitake, T. Polym. J. (Tokyo) 1984,16,187. Higashi, N.; Kunitake, T. Polym. J . (Tokyo) 1984, 16, 583. Kunitake, T.; Tsuge, A,; Nakashima, N. Chem. Lett. 1984, 1783. Kunitake,T.; Higashi, N.; Kajiyama, T. Chem. Lett. 1984,717. Takahara, A.; Kajiyama, T., personal communication. Kunitake, T.; Tawaki, S.; Nakashima, N. Bull. Chem. SOC. Jpn. 1983,56, 3235. Kunitake, T.; Higashi, N. J . Am. Chem. SOC.1985, 107, 692. Kunitake, T.; Asakuma, S.; Higashi, N.; Nakashima, N. Rep. Asahi Glass Found. Ind. Technol. 1984, 45, 163. Okahata, Y.; Ando, R.; Kunitake, T. Ber. Bunsen-Ges. Phys. Chem. 1981,85, 789. Barrer, R. M.; Skirrow, G. J . Polym. Sci. 1948, 3, 549. Hoffman, S. "Depth Profiling" in Practical Surface Analysis; Briggs, D.; Seak, M. P., Eds.; Wiley: New York, 1983. Brundle, C. R. J . VQC.Sci. Technol. 1974, 2 1 , 212. Kajiyama, T.; Nagata, Y.; Washizu, S.; Takayanagi, M. J . Membr. Sci. 1982, 11, 39. Kaiivama. T.: Washizu. S.: Takavanaei. M. J. ADDL '. Polvm. SC;. i984,'29,'3955. Washizu. S.:Terada. I.: Kaiivama. T.: Takavanaei. M. Polvm. ,
,
I
~
"
I
I
I
I
J . ( T O ~ ~ Oi984, ) 16,'307. Kajiyama, T.; Washizu, S.; Ohmori,Y. J . Membr. Sci. 1985,24, 73.
Phase Equilibria in Rodlike Systems with Flexible Side Chains? M. Ballauff Max-Planck-Institut f u r Polymerforschung, 65 Mainz, FRG. Received July 31, 1985
ABSTRACT The lattice theory of the nematic state as given by Flory is extended to systems of rigid rods of axial ratio x and appended side chains characterized by the product of z , the number of side chains per rod, and m, the number of segments per side chain. Soft intermolecular forces are included by using the familiar interaction parameter. Special attention is paid to the form of the orientational distribution function, which is introduced in terms of the refined treatment devised by Flory and Ronca as well as in its approximative form given by Flory in 1956. At nearly athermal conditions the theory presented herein predicts a narrowing of the biphasic gap with increasing volume fraction of the side chains. The region where two anisotropic phases may coexist becomes very small in the presence of side chains. It is expected to vanish at a certain critical volume ratio of side chains and rigid core of the polymer molecule. The dense anisotropic phase at equilibrium with a dilute isotropic phase at strong interactions between the solute particles (widebiphasic region) is predicted to become less concentrated with increase of the product of z and m. The deductions of the treatment presented herein compare favorably with results obtained on lyotropic solutions of helical polypeptides like poly(?-benzyl glutamate). During recent years there has been a steadily growing interest in thermotropic and lyotropic polymers. Spinning or injection molding in the liquid-crystalline state can lead to fibers of high strength and stiffness.' However, the melting point of typical aromatic polyesters exhibiting a mesophase often exceeds 500 O C Y 2 which makes the processing of these materials by conventional methods very difficult. In order to reduce the melting temperature of thermotropic polymers, flexible spacers have been inserted between the mesogenic units.3 But these semiflexible polymers no longer possess the rigidity necessary to produce the desirable mechanical proper tie^.^ Alternatively, sufficient solubility and lower melting point may be Dedicated to the memory of Professor P. J. Flory.
achieved by appending flexible side chains to rigid rod polymers. This third type of liquid-crystalline polymer has been the subject of a number of recent studies. Lenz and co-workers showed that substitution with linear or branched alkyl chain^^,^ considerably lowers the melting point of poly(pheny1ene terephthalate). Similar observations have been made in the course of a study of poly(3n-alkyl-4-hydroxybenzoicacids).' Gray and co-workers demonstrated that side-chain-modified cellulose derivatives exhibit thermotropic as well as lyotropic b e h a ~ i o r . ~ , ~ The solution properties of poly(y-benzyl glutamates) (PBLG) are directly influenced by the presence of flexible side chains, as has been pointed out by Flory and Leonard.'O The thermodynamics of PBLG in various solvents has been investigated theoretically as well as experimentally by Miller and co-workers,"J2 leading to the conclusion
0024-9297/86/2219-1366$01.50/0 0 1986 American Chemical Society
Macromolecules, Vol. 19, No. 5, 1986
--_ A-
Figure 1. Representation of a rigid rod with appended side chains in a cubic lattice.
that the presence of flexible side chains in this polymer profoundly alters the phase diagrams as compared to the rigid rod model. In this work we present a comprehensive treatment of the thermodynamics of stiff rods with appended flexible side chains. As in the theory of Wee and Miller,12 it is based on the lattice model originated by Flory,l3J4which previously has been shown to account for all experimental facts obtained from suitable model syst e m ~ . ~ ~Special - ' ~ attention is paid to the form of the orientational distribution function, as expressed in terms of the Flory-Ronca theorym21or in the 1956 approximative treatment.13 In order to treat lyotropic systems soft intermolecular forces are included by means of the familiar interaction parameter. Steric constraints between adjacent side chains appended to the same rod, an important problem in the course of a statistical-mechanical treatment of membranes22and semicrystalline polymers,23are not taken into account in the present treatment. It is assumed that the distances between neighboring side chains are sufficiently large, as is the case in the polyester systems mentioned a b o ~ e . However, ~!~ as discussed by Flory and Leonard,lo this effect may come into play when helical polypeptides such as PBLG are considered.
Theory The system considered here consists of rodlike molecules with flexible side chains appended regularly (cf. Figure 1). As is customary, we subdivide the lattice into cells of linear dimensions equal to the diameter of the rodlike part of the particle. Thus the number of segments comprising this part is identical with its axial ratio, henceforth denoted by x. For simplicity, we take the segments of the side chains and the molecules of the solvent to be equal in size to a segment of the rodlike part. The axial ratio of the solvent therefore is given by xsolvent= 1. To each rod z side chains are appended comprising m segments. Since the side chains are not incorporated in the main chain, they are not expected to take part in the course of the ordering transitions. Therefore their configuration may be assumed to remain unaffected by the nature of the respective phase. Consider a phase or liquid-crystalline domain where the rodlike parts of the molecules are preferentially oriented to the axis of the domain, the latter being taken along one of the principal axes of the lattice. As in ref 13, the stiff part consists of y sequences containing x / y segments (cf. Figure 1). The relation of y to the angle ic, of inclination to the axis of the domain will be discussed later in this section. The calculation of the configurational partition function rests on the evaluation of the number vj+l of situations available to the j + 1molecule after j particles are introduced. Let nodenote the total number of lattice sites and f = x + mz the overall number of segments per molecule. Pursuant to the calculation of v ~ +it~is, expedient to introduce the rigid part first. Doing this we proceed along
Rodlike Systems with Flexible Side Chains 1367 the lines given in ref 13. The number of situations available to the first segment of the first submolecule thus follows as the number of vacancies no- f j . If there is no correlation between the adjacent rows parallel to the axis of the domain, the expectation of a vacancy required for the yj+l - 1 first segments of the remaining submolecules is the volume fraction of the vacancies (no- f j ) / n o . If a given site for the first segment of a submolecule is vacant, the site following it in the same row can then be only occupied by either the initial segment of a submolecule or a segment of a side chain. The conditional probability therefore is given by the ratio of vacancies to the sum of vacancies, submolecules, and total number of the segments of the side chains (no- fj)/(no- f j + C J y j+ zmj). Inasmuch as we assume that there is no correlation of the conformation of the side chains with the orientational order (see above), the expectancy of a vacancy for the zm segments of the side chains is given by the volume fraction of empty sites (no- fj)/no. It follows that
(y)m (1)
The results deduced by the lattice treatment are expected to become less secure a t greater disorientations, i.e., at larger values of y / x . For the ordered phase under consideration here, this problem should be of minor importance, however. As usual,mthe configurational partition function Z M may be represented as the product of the combinatorial part z W m b , the orientational part Zorient, and the intramolecular or conformational part Zconf: ZM
= ZcombZorientZconf
(4)
Since the configuration of the side chains is assumed to be independent of the order of the respective phase (see above), Zconfmay be dismissed in the subsequent treatment. The combinatorial part is related to v j by
and thus follows as (no- n,x + Yn,)! non,(l-wm) Zcomb = n,!nl!
(6)
where nl = no- fn, denotes the number of solvent particles and n, the number of polymer molecules. As usual, 9 is the average value of the disorder index y defined by (7)
with n,. being the number of molecules whose rigid core is characterized with regard to orientation by y. Following
Macromolecules, Vol. 19, No. 5, 1986
1368 BallaQff
becomes Flory and Ronca,20the orientational part of Zorient
.( T ) uwYnx
Zorient
=
nxy
of the solute particle, and 3 to the flexible side chains. The quantities xL;are given as by xi;
Y
As in ref 20, wy denotes the a priori probability for the interval of orientation related to y, and u is a constant. Introducing Stirling’s approximation for the factorials, eq 4 together with eq 5 and 6 gives -In 2, = n , In u1 + n, In
u, -
= Z,Awi;x;/RT
(17)
where xi represents the number of segments per interacting species, 2, is the coordination number, and Awij (=wi, 0.5(wii + w;;)) denotes the change in energy for the formation of an unlike contact. Since the volume fraction us of the side chains depends on u, through us = u,zm/x, we have
MM= xlnlu, + x2nxu,
(18)
with
and where the quantities u1 = n l / n oand v, = n,x/no denote the volume fractions of the solvent and of the rigid core of the polymer, respectively. For evaluating the orientational distribution we take the last molecule inserted into the lattice as a probe for the orientational order a t equilibrium (see ref 20). Hence, equating j + 1 in eq 2 to n,, we obtain
zm
(20)
XZ = - x 2 3 X
The chemical potentials are derived from eq 7 and 18 with the stipulation that at orientational equilibrium
Hence A p l ’ / R T = In u1 + us’
+ ~u,’
( - 1) y
+a +
which immediately leads to with the quantity a defined by
a = -1n
[
1 - u,( 1 -
91
and (12) X
and
fl = C w , exp(-ay)
(13)
Y
Alternatively, eq 11 may be derived by variational methods.20 Note that the orientational distribution function nxY/nxis given by the same expression as deduced in ref 20. Thus the functional dependence of n x y / n ron the disorientation index y of the rodlike part remains unaffected by the presence of flexible side chains. Accordingly, Zorient follows as Zorient
= nx[ln
(fig)
+ ajil
In what is to follow all quantities referring to the anisotropic phase are marked by a prime. The corresponding expressions for the isotropic phase, obtained by equating y to x , are
(14)
and, upon insertion into (9) u,
A p , / R T = In -
u,
-In Z = nl In u1 + n, In - - (no- n,x) In u,(
1-
:) ]
X
X
+f +f
L
- n,(l - 7 - zm) - n, In ( f l u ) (15)
For examining the effect of small interactions between the solute particles, the heat of mixing A H M is incorporated into the expression for the free energy. If it is assumed that randomness of mixing is preserved a t all degrees of orientation, A H M may be taken to be proportional to the number of segments of one type and the volume fraction of the other species. Hence, introducing the familiar interaction parameters x l z , ~ 1 3 and , ~ 2 3A , H M becomes mM = X12n1u2 + x 1 3 n l u s + X23n2u3 (16) where the index 1refers to the solvent, 2 to the rigid part
For evaluating the composition of the conjugated phases by means of the equilibrium conditions APl = APl’
(25)
AP2 = AP2‘
(26)
the relation between y and the angle of inclination has to be specified. Since the orientational distribution as represented by eq 11does not change when flexible side chains are appended to the stiff rods, we may directly use the expression given by the Flory-Ronca treatment of the lattice model.20 Thus the disorder index y is represented by
Rodlike Systems with Flexible Side Chains 1369
Macromolecules, Vol. 19, No. 5, 1986
In orientational equilibrium its average value follows as
4 9 = ;xfz/f1 and the order parameter s becomes with the integrals f p being given by fp
=
J"'z sinp + exp(-a
sin +) d+
(30)
The factor a relates to a (cf. eq 12) by
4
a = -ax ?r
For values of a >> r/2,the integrals f p may be developed in series, yielding fl
=a +
o(
5) (33)
f3
=
,4 + o ( ; )
(34)
In this asymptotic approximation20the average disorder parameter 9 directly follows from
2
2
9 = a- = In [l - v,(l
-g/x)]
(35)
Thus, we obtain for the chemical potentials eq 23 and 24 the expressions
Apl'/RT = In ul'
u,' + u,' + -@ X
2
- 1) + I + Y
Ap,'/RT =
where C = 2 In ( r e / @ . Further simplification can be achieved by omitting the small constant C and equating u to x (1956approximation; cf. ref 11). In order to obtain the volume fraction of the polymer in the coexisting phases, trial values of u,' and a (cf. eq 12) are chosen which serve for the numerical evaluation of the integrals f p through eq 30. If the quantity a (eq 31) exceeds 30,the f p may be calculated by the series expansion eq 32-34 using the first four terms. With the f p being known, 9 follows from (28), which in turn leads to an improved value of a. The whole calculation is repeated until self-consistency is reached. Equation 26 is then solved numerically to obtain u,, from which an improved value of v i is accessible via eq 25. The entire calculation is repeated with a new value of u,' until compliance with all conditions for equilibrium is reached. When the 1956 approximation is used, the same scheme
applies to eq 25 and 26,with eq 21 and 22 being replaced by eq 36 and 37. Here, self-consistency directly follows from eq 35. The volume fractions u,' and u," of two anisotropic phases in equilibrium can be evaluated by a similar numerical solution of (25)and (26)after insertion of (21)and (22)together with (28). Since this calculation is more sensitive toward the orientational distribution function, the exact treatment of Zorient has to be used.
Results and Discussion Parts a-d of Figure 2 show the phase diagrams resulting for rigid rods of axial ratio x = 100 with flexible side chains characterized by the number of side chains z per rod and the number of segments m per side chain. From the theoretical treatment it is evident that only the product z m matters. Thus the polymer molecules under consideration here are fully determined by x and zm. Since the segments of the side chains are assumed to be isodiametric with the segments of the rigid core, the quantity zmlx directly gives the ratio of the volumes occupied by the respective parts of the molecule. For the present purpose of a discussion of the influence exerted by the side chains, the parameter x1 (see eq 19)measuring the interaction of the solvent and the rigid core will suffice to take into account intermolecular forces. The second parameter xzrelated to the interactions of the side chains with the rigid core only matters at very high concentrations and is thus of minor importance. Figure 2a corresponds to a system of rigid rods immersed in a solvent. The dashed lines indicate the metastable continuations of the phase lines. This diagram has been calculated by Flory in the famous work13 in 1956 using the approximate form of the orientational distribution (see eq 36-38). The result given herein was obtained by resort to the treatment of the lattice model originated by Flory and Ronca.20 A comparison of both methods shows that the main features of the diagram are rather insensitive toward the respective form of the orientational distribution function nXy/n,. However, the stability of the equilibria between two nematic phases is profoundly influenced by nxy/nx.This fact becomes even more important when going to lower axial ratios x . For example, the Flory-Ronca treatment predicts a triphasic equilibrium already for x = 50, an axial ratio not being sufficient to produce a stable nematic-nematic equilibrium in the frame of the 1956 approximation. From these results it is obvious that the exact treatment of n, In, is necessary for the precise calculation of the tripiasic equilibria but is sufficiently accurate for evaluating the isotropic-nematic equilibrium a t axial ratios greater than 100. As can be seen from a comparison of Figure 2, part a, with parts b-d, there is a marked influence of the side chains on the phase behavior of rigid rods. We first discuss Figure 2,part a, 2a together with part b referring to zm = 0 and zm = 2,respectively. Here u p and upl denote the total volume fractions of the polymer in the isotropic phase and in the coexisting ordered phase. Both phase diagrams exhibit the same gross features. A t x1 = 0 the soft interactions between the rods and the solvent are null, corresponding to athermal conditions. The narrow biphasic region at this point is continued until a pair of reentrant nematic phases appears a t a critical point noted on the right-hand side of the ordinate. At a given value of xl,in the regime just beyond the critical point, either of two pairs of phases n,i or n,n may occur. Further increase of xl, equivalent to a lowering of temperature, leads to a triple point where three phases are in equilibrium. Raising x1 to still higher values is followed by a dilute isotropic phase in equilibrium with a dense nematic phase.
1370 Ballauff
Macromolecules, Vol. 19, No. 5, 1986
?, I:
i ' I IN
00
02
A 06 08 1C
OL
0 2 V
X,
T
V "
00
N *
N
8
I
,
02
OL
06
08
02
OL
06
08
IO
~
01
OW1i
N
x
= 100
im
=2
007-
005
I
01
33
05
07
IO VP
0.2
,
00
,
, 10 V,
Figure 3. (a) Volume fraction up and up/ of the polymer in the coexistingisotropicand anisotropic phase, respectively, at athermal conditions vs. the volume fraction u,' of side chains in the anisotropic phase. (b) Volume fraction u, and u,' of the polymer in the coexisting isotropic and anisotropic phase, respectively, at athermal conditions vs. the volume fraction u,' of side chains in the antisotropic phase.
O
' I
'1
- - k - T - F - F + O v.
VP
Figure 2. Phase diagrams of rigid rods immersed in a solvent; x is the axial ratio of the rigid core, z is the number of side chains per rigid core, and m is the number of segments per side chain. The solvent power is measured by the interaction parameter xl. The quantities up and u, denote the volume fraction of the total polymer and its rigid core, respectively. Several features command attention when Figure 2, parts a and b, are compared. Going from zm = 0 (perfectly rigid rods) to zm = 2 narrows considerably the region where to nematic phases are coexisting. If zm > 4,this region be-
comes unstable for x: = 100. Note that both the critical value of x1and its value at the triple point are increasing with appending of side chains. Furthermore, the anisotropic phase coexisting with the isotropic phase above the critical point is less concentrated when compared to zm = 0. Thus, even a small volume fraction of side chains as expressed by the ratio of zm to x suffices to change the phase behavior profoundly. This is even more obvious when higher values of zm (Figure 2c,d) are used. Here, the triphasic equilibrium is no longer stable. In addition, the widening of the biphasic gap occurs at much higher values of xl, as compared to rods without side chains (cf. Figure 2a; note the different scales of the ordinate in Figure 2a-d). Hence, the wide biphasic regime is expected to appear a t a much lower temperature. Another feature of interest is the narrowing of the two-phase region with increase of zm at constant x . For athermal conditions this is shown in more detail in Figure 3a. Here the volume fractions up and upl in the respective phases are plotted against the volume fraction u,' of the side chains in the anisotropic phase. It is evident that appending of even short chains results in a strong narrowing of the biphasic gap. This effect is most pronounced at higher axial ratios x where the two-phase region becomes negligible if u,' > vi. Figure 3b shows the respective volume fractions u, and u,' of the rigid core of the polymer. As is obvious from this plot, the presence of flexible side chains changes the width of the biphasic gap but leaves the magnitude of u, nearly unaffected. Clearly, as in Figure
Macromolecules, Vol. 19, No. 5, 1986
Rodlike Systems with Flexible Side Chains 1371 1-
rv;,
IO1
0
I
10
50
30
x
1M)
Figure 6. Maximum volume fraction of the side chains in the anisotropic phase at up. = 1, i.e., in the neat liquid at athermal
conditions. The quantity xcrit denotes the athermal limit being the minimum axial ratio to produce a stable nematic phase in the absence of attractive forces (cf. ref 20).
'SO i zm -
3
2
1
X
Figure 4. Volume fraction up of the polymer in the isotropic phase at equilibrium at athermal conditions vs. z m / x which gives the
ratio of the volumes occupied by the side chains and the rigid core in the molecule, respectively.
i
1 I
x.100 I
I
1
1
2
3
-
n X
00'
,
1
10
I
,
I
50
I
,
I
,
, 100
c
Figure 5. Volume fraction up of the polymer in the isotropic phase at equilibrium under athermal conditions vs. n, the axial ratio
of the rigid core.
3a, the difference between v, and u,' is vanishing when up = u, + u, is going to unity, i.e., when the whole system is filled with polymer. Since a t athermal conditions u, is nearly independent of zm, the total volume fraction of polymer v in the isotropic phase should increase linearly with zm. fiigwe 4 shows the dependence of up on n(zm/x) to be linear indeed. It has to be noted that up must not be confused with up*, the volume fraction for incipience of metastable order.14 Thus, the ordering process among the rodlike cores is nearly unaffected by the presence of side chains, which appear to act like a low molecular weight solvent. This result clearly relates to the fact that one end of the flexible chain is fixed at the rigid core, which greatly reduces its entropy in solution. At higher polymer concentrations the number of configurations available to the side chains is becoming smaller. As a consequence of this unfavorable entropic effect, the concentration of the polymer in the anisotropic phase tends to smaller values a t increasing values of zm. Since the isotropic phase is located a t a more dilute regime, the overlap between the side chains should be smaller there. Hence, the volume fraction v, of the rigid cores does not change very much upon appending of side chains. Consequently, the curves
Figure 7. Order parameters (cf. eq 29) at different axial ratios x of the rigid core vs. z m l x denoting the ratio of the volume occupied by the flexible side chains and the rigid core in the
molecule, respectively.
describing the dependence of up on x are running nearly parallel to each other, the deviations at lower values of the axial ratio x notwithstanding (cf. Figure 5.) The situation, of course, is changed drastically if the side chains are allowed to move freely. As has been demonstrated by Flory?5 this leads to a marked segregation of the rods and the random coils between the ordered and the isotropic phase. As already discussed above, up reaches unity at a certain volume fraction of side chains (us)". This quantity, which may be taken as a measure of the maximum volume fraction to be tolerated by a nematic phase, is plotted against x in Figure 6. In the limit of high axial ratios x , (uB)-tends to become unity. At low x (us)- is zero at xcrit = 6.417, which is the minimum axial ratio necessary for the occurrence of a nematic phase in the neat liquid.% In the vicinity of xCritthe dependence of (u,)- on x is rather steep, indicating that an ordered phase composed of small rods can only tolerate a small volume fraction of side chains. The foregoing considerations only hold true for athermal conditions. If x1 increases, the biphasic gap is widened, as is obvious from Figure 2b-d. Especially if zm is of the order of x , up becomes very small (cf. Figure 2d). However, the dense nematic phase a t high values of x1 is less concentrated in the presence of side chains than compared to perfectly rigid rods. This is due to the entropic repulsion
1372 Ballauff
Macromolecules, Vol. 19, No. 5, 1986
1
00
02
I
0
I
04
"I
06
Figure 8. Reduced vapor pressure p / p o vs. volume fraction of solvent for an axial ratio of the rigid core x = 100. The quantity zm is the product of z, the number of side chains per rigid core, and m, the number of segments per side chains.
of the flexible side chains discussed above. All results mentioned above suggest that the side chains may be compared to an ordinary solvent to a certain extent. However, as is demonstrated in Figure 7, the order parameter in the nematic phase (cf. eq 29) is lowered with increasing zm. Hence, as opposed to an ordinary solvent consisting of one segment, the flexible side chains interfere with the ordering process of the rodlike core. Finally, we discuss the alterations effected by the presence of side chains a t very high polymer concentrations. For this purpose the reduced vapor pressure p I p o of rods of axial ratio 100 with side chains characterized by zm is plotted vs. the volume fraction of solvent in Figure 8 (athermal conditions). The concentration range is restricted to those within in which the system is homogeneous, the single phase being anisotropic. Even a t small ratios zm/x the resulting p / p odeviates strongly from its respective value a t zm = 0. With zm of the order of x the reduced vapor pressure is fully governed by the side chains. If zm is becoming larger and larger (zm > x ) , the results of the present treatment coincide with those obtained by Flory and Leonard within the frame of their side-chainmixing model.1° Having explored the various consequences of side chains on the ordering transition of rigid rods, we now turn to a comparison with experimental results. As pointed out in the beginning, stiff polyesters with appended flexible alkyl chains5,' seem to be the most promising systems for a quantitative test of the model developed herein. Unfortunately, the few data available on these polyesters only refer to the thermotropic behavior. At present, certain polypeptides exhibiting a stiff helical core in suitable solvents appear to be the best candidates for a test of theory. The cholesteric structure of these solutions does not hamper a comparison with theory since the pitch is much larger than typical molecular dimensions. To the author's knowledge Flory and LeonardlO were the first to point out that the solution behavior of a number of helical polypeptides like poly(ybenzy1 glutamate) stems from the presence of flexible side chains. The thermodynamics of these polymers has been the subject of numerous studies
since, especially by W. G. Miller and co-workers (ref 11, 12,26, 27, and further literature cited therein). Presently, a number of excellent reviews on the subject are available.11828-31Thus it suffices to focus only on the features relevant for the present purpose. For a comparison of theory with experimentaldata one has to bear in mind that the present model only treats molecularly uniform systems. However, the deviations due to the finite width of the distribution of chain length is expected to be of minor importance in the course of a more qualitative discussion. In the phase diagrams predicted by theory three regions have to be distinguished: (1) the narrow biphasic region where x1 is small; (2) the small region in which two anisotropic phases may coexist; and (3) the wide biphasic regime at strong interactions between the rodlike cores. Since the classical work of Robinson et al.32the point in region 1 where the polypeptide solution separates into two phases has been studied exhaustively (see the reviews on the subject, ref 11 and 28-32). This fraction up is accessible through optical33and v i s ~ o m e t r i cmeans. ~~ It has been used repeatedly for testing theoretical models (see, for instance ref 35). However, as has been revealed previously by the careful measurements of S a k a m ~ t oup, ~in~ the isotropic phase strongly depends on solvent power. Due to aggregation PBGL (M,= 1.23 X lo5)exhibits a up of 0.073 in chloroform, whereas dissolving the same polymer in DMF leads to up = 0.124. Recent measurements on light scattering by SchmidP7have demonstrated that DMF is a good solvent for PBGL. Thus aggregation, which clearly leads to a higher axial ratio, should be excluded under these conditions. However, even in DMF the volume fraction in the isotropic phase depends on temperature, as is obvious from the phase diagram given by Miller and co-workers.ll This experimental finding, which may be due to the finite flexibility of the helical makes a comparison of theory and experiment very difficult. A further problem encountered in the experimental investigation of the PBGL/DMF system is its extreme sensitivity toward small amounts of water26that can alter the phase behavior considerably. It is nevertheless interesting to compare the predictions for up by the rigid rod model14 and the theory presented in this paper. Assuming a rodlike structure, Flory showed14that the axial ratios are obtained best from the densities of the respective polymers. Hence, according to this calculation a PBGL of molecular weight 2.20 X lo5 (cf. Table 2 of ref 14) may be characterized by an axial ratio x of 97. From this value of x the volume fraction up in the isotropic phase is given by 0.078. Assuming that the volume of the solid polymer consists of equal parts of rigid core and side chains ( z m / x = 1) leads to an increase of x by a factor of 2lI2; i.e., the axial ratio of 137 thus calculated is followed by up = 0.1322. The present datal' suggest a ratio zm/x smaller than 1, but the temperature dependence of up exhibited by PBGL/DMF and poly[ (carbobenzoxy)lysine]/DMF" does not allow a more quantitative conclusion at the present stage. However, both systems reveal a biphasic gap in region l much smaller than that calculated from the rigid rod model.ll Thus the prediction of a narrowing of the two-phase region (cf. Figure 2a-d) is substantiated by experimental findings. If the solvent power is slightly deteriorated, the rigid rod model predicts the occurrence of two anisotropic phases in equilibrium (see Figure 2a). In the presence of side chains the interaction parameter x1at the point where the systems enters region 2 is shifted to much higher values (cf. Figure 2b). If z m / x exceeds a certain critical value for a given axial ratio, region 2 is predicted to vanish. The experimental investigation of this region is particularly
Macromolecules, Vol. 19, No. 5, 1986
Rodlike Systems with Flexible Side Chains 1373
temperature, and molecular weight will clarify this point. difficult. In most solvents gelation will occur28~38 upon lowering of the temperature, which clearly demonstrates Conclusions the strong interactions between the particles. Even in good It has been shown that the Flory lattice model can be solvents, extreme care has to be taken in order to avoid modified easily to take into account the presence of flexible the occurrence of a "complex phase" due to traces of side chains. The resulting phase diagrams are in qualiwater.26 Nakajima, Hayashi, and O h m ~ r have i ~ ~ given tative agreement with experimental data of lyotropic soevidence for a triphasic equilibrium in the system lutions of helical polypeptides. However, the effect of PBGL/DMF/methanol. Recently, Russo and Miller2' special inter- and intramolecular interactions among the investigated this feature in great detail. From their side chains, as evident in solutions of PBGL, needs further measurements they concluded that PBGL/DMF exhibits exploration, preferably by NMR techniques.& Also, results an equilibrium of two anisotropic phases, whereas polyobtained on systems with varying lengths of side chains46 [ (carbobenzoxy)lysine]/DMF does not show any evidence will clearly lead to a more conclusive test of the theory of such a phenomenon. For comparison of theory and presented in this paper. experiment one has to bear in mind that the rigid rod model predicts region 2 to appear a t small positive values Acknowledgment. The author is indebted to the late of xl. Measurements of the solvent activities by Kubo and Prof. P. J. Flory for helpful discussions and encourage0gin0,~Ohowever, inevitably lead to the conclusion that ment. the small biphasic region 1 is extended to much lower solvent power. Another feature important for the disReferences and Notes cussion of this point is the conformation of the side chains, Prevorsek, D. C. In Polymer Liquid Crystals Cifferi, A., which has been the subject of a number of recent studies Krigbaum, W. R., Meyer, R. B., Eds.; Academic: New York, 1982. by NMR41-43and by measurements of the magnetic susJackson, W. J. Br. Polym. J. 1980, 12, 154. ceptibility.44 The results obtained by these investigations, Ober, Ch. K.; Jin, J.-I.; Lenz, R. W. Adu. Polym. Sci. 1984,59, especially the work by Toriumi, Matsuzawa, and Uemat103. S U , ~demonstrate ~ a finite stiffness of the side chains. Kriebaum. W. R.: Cifferi. A.: Acierno. D. J . A-. D D Polvm. ~. Sci.. Appl. Polym. Symp. 1985,41, 293. Consequently, in the concentration regime under considMainusz, J.: Catala, J. M.: Lenz, R. W. Eur. Polvm. J. 1983. eration here there is a finite order parameter of this moiety 19,-1043. (ca. 0.1) which decreases with increasing t e m p e r a t ~ r e . ~ ~ Dicke, H.-R.; Lenz, R. W. J. Polym. Sci., Polym. Chem. Ed. Hence, a part of the side chains enlarges the rigid core, 1983,21, 2581. Ballauff, M.; Stern, R.; Wegner, G., unpublished data. leading to a lower value of the ratio of zm to x . Similar Tseng, S. L.; Laivins, G. V.; Gray, D. G. Macromolecules 1982, conclusions may be drawn from the comparatively small 15, 1262. values of up in region 1 (see above). Thus, these experiLaivins, G. V.; Gray, D. G. Macromolecules 1985, 18, 1753. mental findings are indicative of an increasing stiffness of Flory, P. J.; Leonard, W. J., Jr. J . Am. Chem. SOC.1965,87, 2102. the side chains in PBLG with ascending concentration. Miller, W. G.; Wu, C . C.; Wee, E. L.; Santee, G. L.; Rai, Y.; However, the system poly[(carbobenzoxy)lysine]/DMF," Goebel, K. G. Pure Appl. Chem. 1974, 38, 37. in which the side chains are longer than in PBGL, does Wee, E. L.; Miller, W. G. Liq. Cryst. Ordered Fluids 1978,3, not show any evidence of a nematie-nematic equilibrium, 371. Flory, P. J. Proc. R. SOC.London, A 1956, 73, 234. in accord with the theory presented herein. Flory, P. J. Adv. Polym. Sci. 1984, 59, 1. Only a few data are available for a discussion of region Irvine, P. A.; Flory, P. J. Wu, Dacheng J. Chem. SOC.,Faraday 3, the wide biphasic gap. In most cases studied so far,28 Trans. 1 1984,80, 1795. Flory, P. J.; Irvine, P. A. J.Chem. SOC.,Faraday Trans. 1 1984, the solutions form a mechanically self-supporting gel when 80, 1807. brought to temperatures near this regime. This, of course, Irvine, P. A,; Flory, P. J. J. Chem. SOC.,Faraday Trans. 1 1984, is in accord with theory, which predicts the widening of 80, 1821. the two-phase region at a much lower solvent power as Ballauff, M.; Wu, Dacheng; Flory, P. J.; Barrall, E. M., I1 Ber. Bunsen-Ges. Phys. Chem. 1984,88, 524. compared to systems consisting of perfectly rigid rods. Ballauff, M.; Flory, P. J. Ber. Bunsen-Ges. Phys. Chem. 1984, Thus gelation may interfere with the ordering process. At 88, 530. the lowest temperatures accessible for PCBL/DMF only Flory, P. J.; Ronca, G. Mol. Cryst. Liq. Cryst. 1979,54, 289. a minute enlargement of the biphasic gap is observed." Flory, P. J.; Ronca, G.; Mol. Cryst. Liq. Cryst. 1979,54, 311. Dill, K. A.; Flory, P. J. Proc. Natl. Acad. Sci. U.S.A. 77, 1980, In PBGL/DMF the existing data on region 311 show the 3115. volume fraction of the dense nematic phase to be smaller Flory, P. J.; Yoon, D. Y . ;Dill, K. A. Macromolecules 1984,17, than that calculated from the rigid rod modell' but wider 862. than expected from the treatment given in this paper. This Flory, P. J. Principles of Polymer Chemistry; Cornel1 University: Ithaca, NY, 1953. again may be due to alterations of the conformation of the Flory, P. J. Macromolecules 1978, 11, 1138. side chains discussed above. Russo, P. S.; Miller, W. G. Macromolecules 1984, 17, 1324. A further aspect of lyotropic solutions that merits atRusso, P. S.; Miller, W. G. Macromolecules 1983, 16, 1690. Uematsu, I.; Uematsu, Y. Adu. Polym. Sci. 1984, 59, 37. tention is the degree of order in the nematic phase. The DuPre, D. B.; Samulski, E. T. In Liquid Crystals, The Fourth lattice theory of rigid rods14 as well as the theory given State of Matter; Saeva, F. D., Ed.; Dekker: New York, 1979. herein makes explicit predictions in this regard in terms Samulski, E. T. In Liquid Crystalline Order in Polymers; of the disorder index y (eq 7) and the familiar order paBlumstein, A., Ed.; New York, 1978. Iizuka, E. Adu. Polym. Sci. 1976, 20, 79. rameter s (eq 29). Although the latter quantity has been Robinson. C.: Ward. I. C.: Beevers. R. B. Discuss. Faradav SOC. exhaustively studied in low molecular weight nematogens, 1958, 25, '29.' it has been largely ignored in investigations of lyotropic Tovoshima, Y.: Minami, N.; Sukigara, M. Mol. Cryst. Lis. polymer solutions. The data given by Samulski and coCrjst. 1976, 35, 325. Hermans, J., Jr. J. Colloid Sci. 1962, 17, 638. w o r k e r ~on~ PBGL ~ in dioxane show s to be of the order Straley, J. P. Mol. Cryst. Liq. Cryst. 1973, 22,333. of 0.5 a t the transition point. A t higher concentration a Sakamoto, R. Colloid Polym. Sci. 1984, 262, 788. value of approximately 0.74.8 results, in better agreement Schmidt, M. Macromolecules 1984, 17, 553. with the data presented in Figure 7. Clearly, a more deSasaki, S.; Tokama, K.; Uematau, I. Polym. Bull. (Berlin) 1983, tailed investigation of s as function of concentration, 10, 539. '
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(43) Czarniecka, K.; Samulski, E. T. Mol. Cryst. Liq. Cryst. 1981, 63, 205. (44) Toriumi, H.; Matsuzama, K.; Uematsu, I. J. Chem. Phys. 1984, 81. 6085. (45) Spiess, H. W. Adu. Polym. Sci. 1985, 66, 24. (46) Hanabusa, H.; Sato, M.; Shirai, H.; Takemoto, K.; Iizuka, E. J . Polym. Sci., Polym. Lett. Ed. 1984, 22, 559.
Interactions between Surfaces with Adsorbed Polymers: Poor Solvent. 2. Calculations and Comparison with Experiment Kevin Ingersent,+Jacob Klein,* a n d Philip P i n c u d Magdalene College, Cambridge, England, Cavendish Laboratory, Cambridge, England, and Polymer Department, Weizmann Institute of Science, Rehouot 76 100, Israel, and Corporate Laboratories, Exxon Research, Clinton Township, Annandale, N e w Jersey 08801. Received September 16, 1985
ABSTRACT: Analytical expressions, based on a mean-field model and derived in a previous paper (part I: Klein, J.; Pincus, P. A. Macromolecules 1982,15,1129),for the interaction between two parallel plates bearing adsorbed polymer in poor-solvent medium, are solved numerically to yield (a) segmental density profiles of the adsorbed polymer and (b) interaction-energyvs. plate-separation profiles. All parameters appearing in these calcuiations are obtainable from bulk data such as the polymer-solvent phase diagram and adsorbance measurements. The calculated results are critically compared with model experiments on forces between smooth mica surfaces bearing polystyrene in cyclohexane in poor-solvent conditions: the agreement is qualitatively very good and quantitatively fair. Some improvements to the model are suggested.
I. Introduction It has long been known that layers of flexible polymers adsorbed at solid-liquid interfaces may strongly modify the surface-surface forces; this effect is commonly used
to stabilize (or destabilize) colloidal dispersions and is also encountered in many naturally occurring biocolloidal systems. The experimental and theoretical situation up to about 1981 has been comprehensively reviewed by Vincent' and by Vincent and Whittington.2 Theoretical treatment of the problem of interaction between surfaces bearing adsorbed polymer layers is complicated by the number of effects that may contribute; these include osmotic interactions between opposing adsorbed segments, volume-exclusion effects due to the impermeable walls, and the sticking energy of segments adsorbed on the solid surface. It may also be possible for single molecules to span the interplate gap, and such bridging always leads to attraction. In addition, any theoretical models must make assumptions about the state of equilibrium of the adsorbed polymer layers when they are moved relative to each other; as surface diffusion phenomena are often very slow, such assumptions require great care. Over the past few years there have been a number of model experimental studies of the interaction between atomically smooth solid (mica) surfaces bearing adsorbed polymer layers and immersed in liquid media in a variety of solvency condition^.^^ These studies are sufficiently direct to invite quantitative comparison between the experimental results and the predictions of theoretical models; in this way we may hope to get insight into the
* Cavendish Laboratory and Weizmann Institute of Science. Address correspondence to the Weizmann Institute. Magdalene College. Present address: Department of Physics, University of Pennsylvania, Philadelphia, PA. Exxon Research. Present address: Department of Chemical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106.
' *
ways in which such models may be improved and also into the validity of the assumptions (such as state of equilibrium) made in such comparisons. Recent attempts at analysis have fallen into two broad categories. Scheutjens and Fleer' have applied the transfer matrix techniques of Rubins and D i M a r ~ i oand , ~ their treatment accounts for the possible chain conformations, suitable weighted on a quasi-crystalline lattice. In the Cahn-de Gennes approach1@12one minimizes the excess free energy of the adsorbed interacting polymer layers with respect to the segmental density profile + ( z ) at distance z from each surface; de Gennes used this approach to calculate the interaction-distance profile between adsorbed polymer layers in a good-solvent system, using scaling exponents to go beyond Flory-Huggins mean-field theory.l2 In an earlier paper,13hereafter referred to as part 1,Klein and Pincus extended de Gennes' treatmenP to calculate the interaction between irreversibly adsorbed polymer layers below the 9-point, where Flory-Huggins mean field is a good approximation.14J5 The assumption of irreversible adsorption is suggested by experiment3and reflects the fact that even when the sticking energy per monomer is small, the total sticking energy per polymer molecule may greatly exceed its thermal energy kBT. Under these poor-solvent conditions an attractive interaction was predicted13at plate separations greater than about R, (the unperturbed radius of gyration of the polymer) and repulsive forces for narrow gaps. This agreed qualitatively with the results of direct force measurements between polystyrene (PS) layers adsorbed on mica surfaces in cyclohexane (CH). We have now used computational methods to obtain force profiles which can be compared quantitatively with experiment. In the next section there is a summary of the mean-field theory from part 1, leading to coupled integral equations that determine the interfacial energy. The set of coupled equations is solved numerically, as briefly described in section 111, and the results are presented in section IV. In section V these predictions are compared critically with
0024-9297/86/2219-1374$01.50/0 0 1986 American Chemical Society
